Bounds for the Quartic Weyl Sum
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910320558080000 |
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| author | Heath-Brown, D. R. |
| author_facet | Heath-Brown, D. R. |
| contents | We improve the standard Weyl estimate for quartic exponential sums in which the argument is a quadratic irrational. Specifically we show that \[\sum_{n\le N} e(αn^4)\ll_{\ep,α}N^{5/6+\ep}\] for any $\ep>0$ and any quadratic irrational $α\in\R-\Q$. Classically one would have had the exponent $7/8+\ep$ for such $α$. In contrast to the author's earlier work \cite{cubweyl} on cubic Weyl sums (which was conditional on the $abc$-conjecture), we show that the van der Corput $AB$-steps are sufficient for the quartic case, rather than the $BAAB$-process needed for the cubic sum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_14531 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bounds for the Quartic Weyl Sum Heath-Brown, D. R. Number Theory 11L15 We improve the standard Weyl estimate for quartic exponential sums in which the argument is a quadratic irrational. Specifically we show that \[\sum_{n\le N} e(αn^4)\ll_{\ep,α}N^{5/6+\ep}\] for any $\ep>0$ and any quadratic irrational $α\in\R-\Q$. Classically one would have had the exponent $7/8+\ep$ for such $α$. In contrast to the author's earlier work \cite{cubweyl} on cubic Weyl sums (which was conditional on the $abc$-conjecture), we show that the van der Corput $AB$-steps are sufficient for the quartic case, rather than the $BAAB$-process needed for the cubic sum. |
| title | Bounds for the Quartic Weyl Sum |
| topic | Number Theory 11L15 |
| url | https://arxiv.org/abs/2312.14531 |